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8-2020
Optimization of Aileron Spanwise Size and Shape to Minimize Optimization of Aileron Spanwise Size and Shape to Minimize
Induced Drag in Roll with Correlating Adverse Yaw Induced Drag in Roll with Correlating Adverse Yaw
Joshua R. Brincklow Utah State University
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OPTIMIZATION OF AILERON SPANWISE SIZE AND SHAPE TO MINIMIZE
INDUCED DRAG IN ROLL WITH CORRELATING ADVERSE YAW
by
Joshua R. Brincklow
A thesis submitted in partial fulfillment
of the requirements for the degree
of
MASTER OF SCIENCE
in
Mechanical Engineering
Approved:
______________________ ____________________
Douglas Hunsaker, Ph.D. Geordie Richards, Ph.D.
Major Professor Committee Member
______________________ ____________________
David Geller, Ph.D. Richard S. Inouye, Ph.D.
Committee Member Vice Provost for Graduate Studies
UTAH STATE UNIVERSITY
Logan, Utah
2020
iii
ABSTRACT
Optimization of Aileron Spanwise Size and Shape to Minimize Induced Drag in Roll
with Correlating Adverse Yaw
by
Joshua R. Brincklow, Master of Science
Utah State University, 2020
Major Professor: Dr. Douglas Hunsaker, Ph.D.
Department: Mechanical and Aerospace Engineering
Most modern aircraft employ discrete ailerons for roll control. The induced drag,
rolling moment, and yawing moment for an aircraft is dictated in part by the location and
spanwise size of the ailerons. To quantify these forces and moments and relate them to
aileron design, a potential-flow lifting-line theory is used. This work explores a large
design space composed of linearly tapered wing planforms and aileron geometries. Lifting-
line theory shows that the optimum aileron location for minimizing induced drag always
extends to the wing tip. This aileron design is not influenced by lift and rolling moment
requirements. Changes to optimum aileron designs and their impacts to induced drag and
yawing moment are considered and provide context for benefits in future morphing aircraft.
Results are provided to give insight into aileron placement in the early design process. In
most cases, optimum discrete ailerons produce 5β20% more induced drag than a morphing
wing at the same rolling moment.
(46 pages)
iv
PUBLIC ABSTRACT
Optimization of Aileron Spanwise Size and Shape to Minimize Induced Drag in Roll
with Correlating Adverse Yaw
Joshua R. Brincklow
Most modern aircraft make use of modifying the main wing in flight to begin a roll.
In many cases, this is done with a discrete control surface known as an aileron. The lift,
drag, and moments for the wing are affected in part by the location and size of the ailerons
along the length of the wing. The lift, drag, and moments can be found using a lifting-line
theory that considers the circulation in airflow from many small sections of the wing. To
minimize the drag due to lift on the wing, the ailerons must be optimized for the best
location and size. In every case, the optimum aileron size extends to the wing tip. Results
are provided in plots that can be used during the early design process to select optimum
aileron size and location, as well as find the corresponding moments and drag due to lift.
Compared to morphing wings, or wings that can change their shape for a new lift
distribution along the wing, optimum discrete ailerons produce 5β20% more drag due to
lift at the same rolling moment.
vi
ACKNOWLEDGMENTS
This work was funded by the U.S. Office of Naval Research Sea-Based Aviation
program (Grant No. N00014-18-1-2502) with Brian Holm-Hansen as the program officer.
Completion of this work would not be possible without my advisor, as he provided
the guidance necessary to see the bigger picture and sidestep the pitfalls so common in
research.
Joshua R. Brincklow
vii
CONTENT
ABSTRACT ...................................................................................................................... iii
PUBLIC ABSTRACT ...................................................................................................... iv
ACKNOWLEDGMENTS ................................................................................................ vi
LIST OF FIGURES ........................................................................................................ viii
NOMENCLATURE ......................................................................................................... ix
1 INTRODUCTION ...................................................................................................1
1.1 Background .......................................................................................................1
1.2 Lifting-line Theory............................................................................................2
1.3 Optimum Twist Distributions ...........................................................................9
2 LIFTING-LINE ANALYSIS OF ROLL INITIATION ........................................12
2.1 Background .....................................................................................................12
2.2 Derivation of Novel Terms for Aileron Effects ..............................................12
3 APPLICATION OF THE NUMERICAL LIFTING-LINE METHOD.................17
3.1 Comparison of the Classical and Numerical Lifting-line Methods ................17
3.2 Case Setup .......................................................................................................20
3.3 Grid Convergence and Optimization ..............................................................21
4 EMPIRICAL RELATIONS FOR DESIGN BASED ON RESULTS ...................27
4.1 Processing the Results.....................................................................................27
4.2 Results .............................................................................................................27
4.2.1 Optimal Aileron Root ......................................................................27
4.2.2 Values of π π·β ...................................................................................28
4.2.3 Values of π π .....................................................................................29
4.3 Application Example .......................................................................................30
5 CONCLUSION ......................................................................................................32
REFERENCES ..................................................................................................................37
viii
LIST OF FIGURES
Figure Page
1 Change in local section angle of attack due to pure rolling rate .........................4
2 Induced drag planform penalty factor for untwisted linearly tapered
wings ...................................................................................................................9
3 Rectangular planforms with varying methods of spanwise node
placement with a lifting-line along the quarter-chord .......................................18
4 Induced drag increment error between two lifting-line methods as a
function of nodes per semispan. ........................................................................19
5 Grid-convergence analysis of induced drag coefficient at several
prescribed rolling moment coefficients. ............................................................21
6 Aircraft properties as a function of grid density ...............................................22
7 Contour of induced drag coefficient at a rolling-moment coefficient of
0.04 ....................................................................................................................24
8 Contour of induced drag coefficient at a rolling-moment coefficient of
0.1 ......................................................................................................................24
9 Contour plot of deflection angle (in degrees) ...................................................25
10 Contour plot of yawing moment coefficient .....................................................25
11 Aileron root positions based on aspect ratio and taper ratio to achieve
minimum induced drag .....................................................................................28
12 Values of π π·β using optimal aileron design as a function of taper ratio
for aspect ratios ranging from 4 to 20 ...............................................................29
13 Values of π π using optimal aileron design as a function of taper ratio
for aspect ratios ranging from 4 to 20 ...............................................................30
ix
NOMENCLATURE
π΄π = Fourier coefficients in the lifting-line solution
ππ = decomposed Fourier coefficients related to planform
π = semispan of the wing
ππ = decomposed Fourier coefficients related to symmetric twist
πΆπ·π = induced drag coefficient
πΆπ·0 = simplified induced drag coefficient
οΏ½ΜοΏ½πΏ,πΌ = section-lift slope
πΆβ = rolling-moment coefficient
πΆπ = yawing-moment coefficient
π = local section chord length
ππ = decomposed Fourier coefficients related to aileron deflection
ππ = decomposed Fourier coefficients related to rolling rate
οΏ½ΜοΏ½ = local section lift
π = number of terms retained in a truncated infinite series
π = angular rolling rate, positive right wing down
οΏ½Μ οΏ½ = dimensionless angular rolling rate
π π΄ = wing aspect ratio
π π = wing taper ratio
πβ = freestream velocity magnitude
π§ = spanwise coordinate from mid-span, positive left
π§πΏπ = spanwise position of the aileron closest to the wing root
x
π§πΏπ‘ = spanwise position of the aileron closest to the wing tip
πΌ = local geometric angle of attack relative to the freestream
πΌπΏ0 = local zero-lift angle of attack
Ξ = local section circulation
ΞπΆπ·π = incremental change in induced-drag coefficient
πΏπ = aileron deflection angle in radians
πΏπ = semispan position of the aileron closest to the wing root
πΏπ‘ = semispan position of the aileron closest to the wing tip
νπ = local airfoil-section flap effectiveness
νΞ© = twist effectiveness
π = change of variables for the spanwise coordinate
π π· = planform penalty factor in induced drag calculations
π π·πΏ = lift factor in induced drag calculations
π π·β = rolling-moment factor in induced drag calculations
π π·Ξ© = twist factor in induced drag calculations
π π = yawing-moment factor in yawing-moment calculations
π = air density
π = spanwise antisymmetric twist distribution function
Ξ© = negative of the twist value at the location of max magnitude twist
π = spanwise symmetric twist distribution function
CHAPTER 1
INTRODUCTION
1.1 Background
Discrete control surfaces are often used on a main wing for roll control and are
often referred to as ailerons. Aircraft performance, structure, and system configuration,
which vary with each airframe design, often determines the size and placement of the
ailerons [1β4]. In recent years, extensive studies have been made into morphing aircraft
that can deflect the wing trailing-edge continuously. For example, NASA has studied
morphing wing concepts such as the Variable-Camber Continuous Trailing Edge
(VCCTE) [5]. Flexsys is working on a continuous trailing-edge flap [6] for a Gulfstream
aircraft. The Utah State University Aerolab in partnership with AFRL has developed
and flight-tested a variable-camber continuous wing (VCCW) [7β11]. Often the main
benefit of morphing-wing technology over ailerons is minimizing drag for a range of
flight conditions [5,12β14]. Aileron deflection produces increased induced drag and
radar observability, and decreased roll-yaw coupling control compared to morphing
wings at a given rolling moment [15,16].
The location and spanwise size of the ailerons partially determine the magnitude
of the yawing moment and drag. An optimal aileron geometry is desired to compare
against morphing wings, as non-optimal solutions for discrete control surfaces
compared against optimal continuous trailing-edge surfaces will lead to incorrect
conclusions. Optimal aileron geometries are also desired for insight into aileron
placement during the early stages of aircraft design. A potential-flow vortex lattice
analysis for optimal aileron placement was performed by Feifel [17] for elliptic
2
planforms with a rolling moment requirement to minimize induced drag. This work
broadens the scope by considering linearly tapered wings with ailerons using potential-
flow lifting-line theory and gradient optimization. A relationship between aileron
placement, induced drag, rolling moment, and yawing moment is provided as part of
Prandtlβs classical lifting-line theory [18] and gives further insight into the conclusions
made from the numerical results.
1.2 Lifting-line Theory
The section-lift distribution and induced drag on a finite wing is expressed in a
Fourier sine series in Prandtlβs classical lifting-line (LL) theory [18,19]. The classical LL
solution for the circulation distribution can be expressed as
π€(π) = 2ππββπ΄π sin(ππ)
π
π=1
(1)
where b represents the semispan of the wing, πβ represents the freestream velocity, and π
represents a change of variables in the spanwise direction,
π β‘ cosβ1(β2π§/π) (2)
Combining Eq. (1) with the Kutta-Joukowski law [20,21] gives
οΏ½ΜοΏ½(π) = 2ππβ2πβπ΄π sin(ππ)
π
π=1
(3)
The Fourier coefficients in Eqs. (1) and (3) are related to the distributions of the
chord-length and aerodynamic angle-of-attack. Prandtlβs LL equation can be used for any
3
wing planform and twist distribution to determine the spanwise section-lift distribution. To
obtain the Fourier coefficients π΄π in Eqs. (1) and (3), the LL equation must be satisfied at
π locations along the wing. This results in a linear system that can be solved to yield the
Fourier coefficients
βπ΄π [4π
οΏ½ΜοΏ½πΏ,πΌπ(π)+
π
sin(π)] sin(ππ)
π
π=1
= πΌ(π) β πΌπΏ0(π) (4)
where πΌ(π) and πΌπΏ0(π) are functions of spanwise location and represent the geometric and
aerodynamic angle of attack respectively. These can be used once the Fourier coefficients
have been obtained to solve for the integrated forces and moments on the wing. With rigid-
body roll effects included, the resultant lift, induced drag, rolling moment, and yawing
moment coefficients are
πΆπΏ = ππ π΄π΄1 (5)
πΆπ·π = ππ π΄βππ΄π2
π
π=1
βππ π΄οΏ½Μ οΏ½
2π΄2 (6)
πΆβ = βππ π΄4π΄2 (7)
πΆπ =ππ π΄4β(2π β 1)π΄πβ1π΄π
π
π=2
βππ π΄οΏ½Μ οΏ½
8(π΄1 + π΄3) (8)
where οΏ½Μ οΏ½ is the nondimensional roll rate about the stability axis, and is defined as
οΏ½Μ οΏ½ β‘ ππ/2πβ (9)
4
The stability axis is the axis parallel to the freestream and intersecting the center of gravity
as shown in Fig. 1.
Fig. 1 Change in local section angle of attack due to pure rolling rate.
Equations (4)β(8) have the disadvantage of requiring recalculation for each change in
operating condition, including angle of attack, control-surface deflection, and rolling rate.
A more useful form of the LL solution has been presented by Phillips and Snyder [22] and
allows for operating conditions to be solved for independently [23]. A similar approach is
applied here with the definition
π΄π = ππ(πΌ β πΌπΏ0)root β πππΊ + πππΏπνπ + πποΏ½Μ οΏ½ (10)
where ππ , ππ , ππ , ππ are decomposed Fourier coefficients representing planform, twist,
aileron deflection, and roll rate, respectively. Here twist is defined to be spanwise
symmetric with scaling βΞ©, and the roll control mechanism to be symmetric in magnitude
and opposite in sign, termed antisymmetric, with a magnitude of πΏπ. The symbol νπ is the
aileron section flap effectiveness, which in this work is assumed to be constant across the
z
pVβ
βzp
Vβ
zp
Dap
βDap
Vβ
5
span of the flap. The decomposed Fourier coefficients can be found by using the relations
βππ [4π
οΏ½ΜοΏ½πΏ,πΌπ(π)+
π
sin(π)] sin(ππ) = 1
π
π=1
(11)
βππ [4π
οΏ½ΜοΏ½πΏ,πΌπ(π)+
π
sin(π)] sin(ππ)
π
π=1
= π(π) (12)
βππ [4π
οΏ½ΜοΏ½πΏ,πΌπ(π)+
π
sin(π)]
π
π=1
sin(ππ) = π(π) (13)
βππ [4π
οΏ½ΜοΏ½πΏ,πΌπ(π)β
π
sin(π)] sin(ππ)
π
π=1
= cos(π) (14)
where π(π) is a symmetric twist distribution function, and π(π) is a spanwise
antisymmetric twist distribution function, which can be represented as an indicator function
π(π§) =
{
0, π§ < βπ§πΏπ‘1, βπ§πΏπ‘ β€ π§ β€ βπ§πΏπ0, βπ§πΏπ < π§ < π§πΏπβ1, π§πΏπ β€ π§ β€ π§πΏπ‘0, π§ > π§πΏπ‘
(15)
where π§πΏπ is the spanwise position of the aileron closest to the wing root and π§πΏπ‘ is the
spanwise position of the aileron closest to the wing tip. The aileron root and tip are here
defined as the spanwise edge position of the aileron closest to the wing root and tip,
respectively.
The normalized twist distribution functions π(π) and π(π) are multiplied by the
corresponding scalings βΞ© and πΏπνπ to give the resultant total twist distribution in the
6
wing. The total symmetric twist is βΞ©π(π), and the total antisymmetric twist is πΏπνππ(π).
For a given wing planform, symmetric twist distribution function, and antisymmetric
control deflection distribution, Eqs. (11)β(14) can be solved for the decomposed Fourier
coefficients. These coefficients along with angle-of-attack, symmetric twist scaling,
control deflection scaling, section flap effectiveness, and rolling rate can then be used in
Eq. (10) to compute the Fourier coefficients in Eqs. (5)β(8).
For a wing with symmetric planform and twist, the even terms of the ππ and ππ
coefficients are zero. For any wing with an antisymmetric control surface distribution, the
odd terms of the ππ coefficients are zero. The odd terms of the ππ coefficients are also zero,
since the aerodynamic angle-of-attack changes antisymmetrically with rigid-body roll
about the stability axis. Equation (10) can then be expressed as
π΄π = {ππ(πΌ β πΌπΏ0)root β πππΊ j odd
πππΏπνπ + πποΏ½Μ οΏ½ j even (16)
Using Eq. (16) in Eqs. (5)β(8) gives
πΆπΏ = ππ π΄[π1(πΌ β πΌπΏ0)root β π1πΊ] (17)
πΆπ·π = ππ π΄βπ[ππ(πΌ β πΌπΏ0) β πππΊ]2+ ππ π΄βπ(πππΏπνπ + πποΏ½Μ οΏ½)
2π
π=2
π
π=1
βππ π΄οΏ½Μ οΏ½
2(π2πΏπνπ + π2οΏ½Μ οΏ½)
(18)
πΆβ = βππ π΄4(π2πΏπνπ + π2οΏ½Μ οΏ½) (19)
7
πΆπ = βππ π΄οΏ½Μ οΏ½
8(π1(πΌ β πΌπΏ0)root β π1Ξ© + π3(πΌ β πΌπΏ0)root β π3Ξ©)
+ππ π΄4((β(2π β 1)(ππβ1(πΌ β πΌπΏ0)root β ππβ1Ξ©)(πππΏπνπ + πποΏ½Μ οΏ½)
π
π=2
)
j even
+ (β(2π β 1)(ππβ1πΏπνπ + ππβ1οΏ½Μ οΏ½)(ππ(πΌ β πΌπΏ0)root β ππΞ©)
π
π=3
)
j odd
)
(20)
Recognizing the last term in Eq. (18) is the same as Eq. (19), Eq. (18) can be expressed as
πΆπ·π = ππ π΄βπ[ππ(πΌ β πΌπΏ0) β πππΊ]2
π
π=1
+ ππ π΄βπ(πππΏπνπ + πποΏ½Μ οΏ½)2
π
π=2
β 2οΏ½Μ οΏ½πΆβ
(21)
In the absence of aileron deflection and rolling rate, the induced drag simplifies to
πΆπ·0 = ππ π΄βπ[ππ(πΌ β πΌπΏ0) β πππΊ]2
π
π=1
(22)
and the induced drag can be rearranged in the form [24]
πΆπ·0 =πΆπΏ2(1 + π π·) β π π·πΏπΆπΏπΆπΏ,πΌπΊ + π π·πΊ(πΆπΏ,πΌπΊ)
2
ππ π΄ (23)
where
πΆπΏ = πΆπΏ,πΌ[(πΌ β πΌπΏ0)root β νπΊπΊ] (24)
8
πΆπΏ,πΌ = ππ π΄π1 =οΏ½ΜοΏ½πΏ,πΌ
[1 + οΏ½ΜοΏ½πΏ,πΌ/(ππ π΄)](1 + π πΏ) (25)
π πΏ β‘1 β (1 + ππ π΄/οΏ½ΜοΏ½πΏ,πΌ)π1
(1 + ππ π΄/οΏ½ΜοΏ½πΏ,πΌ)π1 (26)
νπΊ β‘π1π1
(27)
π π· β‘βπππ2
π12
π
π=2
(28)
π π·πΏ β‘ 2π1π1βπ
ππ
π1(ππ
π1βππ
π1)
π
π=2
(29)
π π·πΊ β‘ (π1π1)2
βπ(ππ
π1βππ
π1)
2π
π=2
(30)
Here ππ depends on planform as shown in Eq. (11) and ππ depends on wing twist
as shown in Eq. (12), and therefore π πΏ, νΞ©, π π·, π π·πΏ, and π π·Ξ© depend on planform and twist,
with examples shown by Phillips et. al. [25]. From these relationships, valuable
conclusions can be drawn about optimum taper ratio and symmetric twist design. For
example, in the absence of twist, Eq. (23) simplifies to
πΆπ·0 =πΆπΏ2
ππ π΄(1 + π π·) (31)
The term π π· represents the wing planform penalty factor in induced drag relative
to an untwisted elliptic wing. The wing planform penalty factor can be computed for any
planform from Eqs. (28) and (31). Glauert [26] was the first to visualize π π·, and more
9
recently, Phillips [27] produced a similar figure. Work by Phillips et. al [28] cleared up
misconceptions based on Glauertβs limited results. In Fig. 2, a visualization for π π· as a
function of taper ratio and aspect ratio is given, similar to Phillips [27]. The common rule-
of-thumb that induced drag is minimized with a taper ratio of 0.4 comes from these types
of computations [26,27].
Fig. 2 Induced drag planform penalty factor for untwisted linearly tapered wings.
These types of design-space explorations are useful for providing intuition in the
early stages of aircraft design. Here, a similar approach is used to examine the influence of
aileron placement on induced drag. In order to evaluate the effect of discrete ailerons on
induced drag and find the correlated yawing moment, it is helpful to understand two
optimal twist distributions.
1.3 Optimum Twist Distributions
Phillips, et. al [14] and Phillips and Hunsaker [29] showed a normalized spanwise
10
twist distribution function that minimizes induced drag for any symmetric wing planform
in steady level flight. This can be written as
π(π) = 1 βπ ππ(π)
π(π)/πππππ‘ (32)
with the required symmetric twist scaling based on the lift coefficient
πΊ =4ππΆπΏ
ππ π΄οΏ½ΜοΏ½πΏ,πΌπππππ‘ (33)
And the required angle of attack is
(πΌ β πΌπΏ0)ππππ‘ =πΆπΏππ π΄
(4π
οΏ½ΜοΏ½πΏ,πΌπππππ‘+ 1) (34)
Any wing employing this twist distribution function at the angle of attack given in Eq. (34)
will produce an elliptic lift distribution and result in an induced drag of
πΆπ·π =πΆπΏ2
ππ π΄ (35)
Similarly, a LL analysis by Hunsaker et. al [30] produced an antisymmetric twist
distribution function that minimizes induced drag for any symmetric wing planform and
prescribed rolling moment with zero rolling rate. This can be written as
π(π) = [1 +2π π ππ(π)
οΏ½ΜοΏ½πΏ,πΌπ(π)] πππ (π) (36)
This twist distribution function provides the optimum continuous twist along the wingspan
to produce a given rolling moment and minimize induced drag in the absence of rolling
rate. For any wing geometry employing the optimal antisymmetric twist distribution
11
function from Hunsaker et. al [30] in Eq. (36), the corresponding induced drag increase
relative to steady level flight conditions is
(π₯πΆπ·π)β = 32πΆβ2
ππ π΄ (37)
Equation (37) gives the minimum increase in drag for any rolling moment. Most aileron
designs produce more induced drag than this. Using the antisymmetric twist distribution
function given by Eq. (36), the resulting yawing moment at roll initiation is given by
πΆπ = β3πΆπΏπΆβππ π΄
β 5πΆβ(π3(πΌ β πΌπΏ0)ππππ‘ β π3πΊ) (38)
With this LL formulation, the effect of discrete ailerons on induced drag can now be
considered.
12
CHAPTER 2
LIFTING-LINE ANALYSIS OF ROLL INITIATION
2.1 Background
Classical LL theory uses a Fourier sine series to represent the lift distribution along
a wing as shown in Eq. (3). Including more Fourier coefficients increases the number of
frequencies considered in the analysis as well as the rank of the linear system of equations
that must be solved. An aileron deflection represents a step change in twist distribution
along the wing, and therefore introduces many frequencies into the lift distribution.
Because solutions to this system of equations were obtained by hand in the early days of
aeronautics, they were limited in the number of Fourier coefficients that could be included.
Hence, it was difficult to use this method to accurately evaluate the effect of ailerons on
induced drag before the advent of the computer. Today, however, it is quite simple to solve
large systems of equations with little effort, so many more Fourier coefficients can be
included.
2.2 Derivation of Novel Terms for Aileron Effects
The change in induced drag based only on aileron deflection and rolling rate can be
obtained by subtracting Eq. (22) from Eq. (21). This gives
(π₯πΆπ·π)πΏοΏ½Μ οΏ½ β‘ πΆπ·π β πΆπ·0 = ππ π΄βπ(πππΏπνπ + πποΏ½Μ οΏ½)2β 2οΏ½Μ οΏ½πΆβ
π
π=2
(39)
During roll initiation, the rolling rate is zero, while the aileron deflection and rolling
13
moment are nonzero. In this case, the rolling moment, yawing moment, and change in
induced drag can be found from Eqs. (19), (20), and (39) respectively by using οΏ½Μ οΏ½ = 0,
which gives
πΆβ = βππ π΄4π2πΏπνπ (40)
πΆπ =ππ π΄πΏπνπ
4((β(2π β 1)(ππβ1(πΌ β πΌπΏ0)root β ππβ1Ξ©)(ππ)
π
π=2
)
j even
+ (β(2π β 1)(ππβ1)(ππ(πΌ β πΌπΏ0)root β ππΞ©)
π
π=3
)
j odd
)
(41)
(π₯πΆπ·π)πΏ = ππ π΄πΏπ2νπ2βπππ
2
π
π=2
(42)
Using Eq. (40) in Eq. (42) to eliminate the aileron deflection magnitude πΏπ gives
(π₯πΆπ·π)πΏ =32πΆβ
2(1 + π π·β)
ππ π΄ (43)
where
π π·β β‘1
2βπ (
ππ
π2)2
π
π=4
(44)
14
From Eq. (43), it is shown that the increase in induced drag is a function of the aspect ratio,
rolling moment, and π π·β. The decomposed Fourier coefficients ππ depend on planform as
shown in Eq. (13), as well as the spanwise aileron edge positions as shown in Eq. (15).
Hence, the value for π π·β is a function of planform, aileron position, and aileron spanwise
length. Note however that this analysis predicts that the increase in induced drag given in
Eq. (43) is independent of section flap effectiveness νπ, lift, and symmetric twist. It is also
interesting to note that for this case, the increase in induced drag is directly proportional to
the square of the rolling moment, much in the same way that the induced drag in the
absence of twist is proportional to the square of the lift coefficient, as shown in Eq. (31).
The induced drag of an untwisted wing of any planform with ailerons can be given as
πΆπ·π =πΆπΏ2(1 + π π·) + 32πΆβ
2(1 + π π·β)
ππ π΄ (45)
If the symmetric twist distribution function given in Eqs. (32) and (33) is used, π π· is zero.
If the optimal antisymmetric twist distribution function from Hunsaker et. al [30] given in
Eq. (36) is used, π π·β is zero. If both twist distribution functions are used simultaneously,
the minimum induced drag for a given lift and rolling moment is [14]
πΆπ·π =πΆπΏ2 + 32πΆβ
2
ππ π΄ (46)
As a first step to understand aileron design, only wings without twist will be
considered, making Ξ© zero. This is similar to the approach Glauart [26] and Phillips [27]
employed, since they neglected twist in their initial studies on the effects of wing planform
on induced drag. Applying these simplifications to Eqs. (17) and (41) gives
15
πΆπΏ = ππ π΄[π1(πΌ β πΌπΏ0)root] (47)
πΆπ =ππ π΄πΏπνπ
4((β(2π β 1)(ππβ1(πΌ β πΌπΏ0)root)(ππ)
π
π=2
)
j even
+ (β(2π β 1)(ππβ1)(ππ(πΌ β πΌπΏ0)root)
π
π=3
)
j odd
)
(48)
Substituting Eq. (47) in Eq. (48) gives:
πΆπ =πΆπΏπΏπνπ
4((β(2π β 1) (
ππβ1
π1) (ππ)
π
π=2
)
j even
+ (β(2π β 1)(ππβ1) (ππ
π1)
π
π=3
)
j odd
)
(49)
Equation (40) can be rearranged and used in Eq. (49) to eliminate the aileron deflection
magnitude πΏπ:
πΆπ = βπΆπΏπΆβππ π΄
((β(2π β 1) (ππβ1
π1) (ππ
π2)
π
π=2
)
j even
+ (β(2π β 1) (ππβ1
π2) (ππ
π1)
π
π=3
)
j odd
)
(50)
This can be rearranged to give
16
πΆπ = βπΆπΏπΆβπ πππ π΄
(51)
where
π π = 3 + (β(2π β 1) (ππβ1
π2) (ππ
π1)
π
π=3
)
j odd
+ (β(2π β 1) (ππβ1
π1) (ππ
π2)
π
π=4
)
j even
(52)
17
CHAPTER 3
APPLICATION OF THE NUMERICAL LIFTING-LINE METHOD
3.1 Comparison of the Classical and Numerical Lifting-line Methods
A current major drawback of using classical LL theory in the evaluation of the
effects of ailerons on induced drag is the inability to use grid clustering to achieve second-
order convergence. In the classical LL theory, traditionally nodes are clustered along the
wing using Eq. (2) with cosine-clustering near the wing tips by evenly spacing the nodes
in π. However, this method of clustering does not consider how the node clustering will
fall relative to the placement of the aileron. Figure 3(a) provides a visualization of the
traditional cosine-clustering with 80 nodes and symmetrically-placed ailerons. Note that
the edge of the aileron may be in a position between two nodes or directly on a node. As
the number of nodes used in the calculation increases, the accuracy of the induced drag and
yawing moment solutions will vary as the cosine-clustered nodes change position relative
to the location of the edge of the aileron. As a comparison, the clustering used in Fig. 3(b)
allows greater control over the placement of nodes relative to the aileron position, so that
the edge of an aileron falls directly on a node regardless of its span or spanwise location.
This clustering can be termed as aileron-sensitive clustering. Aileron-sensitive clustering
can be utilized in the classical LL theory, but because an aileron forces a step change in
twist along the span, an infinite number of frequencies are introduced, and the finite Fourier
series cannot accurately model the step change in twist. Finding mathematical workarounds
to use the classical LL method with aileron-sensitive clustering is a future topic of study.
Given the prior work in deriving the LL theory, the current inability to access a
working induced drag and yawing moment solution using discrete control surfaces may
18
seem a great discouragement. However, a numerical LL algorithm published by Phillips
and Snyder [22], which is a close numerical analog to the classical LL theory, can
effectively use aileron-sensitive clustering and accurately find the aerodynamic effects of
ailerons.
Fig. 3 Rectangular planforms with varying methods of spanwise node placement with
a lifting-line along the quarter-chord.
The numerical LL method differs from the vortex lattice method that Feifel [17]
used because the surface flow boundary condition is not required at the three-quarter-chord
[31] along the panel center line. Instead, the algorithm finds the local circulation at each
wing section with a relationship between the three-dimensional vortex lifting law [32] and
the section airfoil lift. This algorithm depends on a system of lifting surfaces connected by
discrete horseshoe vortices, creating a vorticity field [33]. This method can be applied to
multiple lifting surfaces with sweep and dihedral and gives accurate solutions for wings
with aspect ratios greater than about 4 [22]. This algorithm is applied in MachUp [14,34],
an open-source code available on GitHub1.
As an example, Fig. 4 shows the difference in convergence between MachUp and
1 https://github.com/usuaero/MachUp
19
the classical LL method with traditional cosine-clustering for an induced drag increment
caused by aileron deflection as a function of nodes per semispan. Error is calculated as the
difference in induced drag increment between a given number of nodes per semispan and
a significantly greater number of nodes per semispan, in this case 640 nodes, using the
same method. Note that the classical LL method does not consistently decrease in error as
the number of nodes is increased. The numerical LL algorithm is therefore a more
consistently accurate tool to explore the design space of aileron sizing and placement.
Fig. 4 Induced drag increment error between two lifting-line methods as a function
of nodes per semispan.
A downside of using the numerical lifting-line algorithm is that the results for the
decomposed Fourier coefficients cannot be found directly. The numerical lifting-line
algorithm provides integrated force and moment solutions for the complete wing, which
can be used to estimate π π·β and π π. Rearranging Eqs. (31), (43), and (51) gives
π π· =πΆπ·0ππ π΄
πΆπΏ2 β 1 (53)
20
π π·β =(ΞπΆπ·π)πΏ(ππ π΄)
32πΆβ2 (54)
π π = βππΆππ π΄πΆπΏπΆβ
(55)
Results for the integrated induced-drag increment and yawing moment were then used in
Eqs. (53)β(55) to estimated π π·, π π·β, and π π for a given planform and aileron geometry.
3.2 Case Setup
Each wing semispan is specified in 3 wing sections, with the center section
containing the aileron, and each wing section is cosine-clustered with a number of nodes
relative to section span, as shown in Fig. 3(b). The control surfaces are modeled with a
flap-chord fraction of 1.0 for the entire control surface length. As Feifel [5] notes, the
induced drag increment predicted by potential flow algorithms is independent of the
aileron flap-chord fraction and depends only on the prescribed rolling moment,
explained in detail by Phillips [15] as well as Abbott and Doenhoff [22]. A Newton-
Secant method is used to find the aileron deflection that would provide a target rolling
moment within machine precision at double-precision computing.
The angle of attack and rolling moment coefficient were adjusted iteratively to
arrive at the prescribed lift coefficient and rolling-moment coefficient for a given wing
planform and aileron geometry. Convergence criteria of 1.0 Γ 10-12 and 1.0 Γ 10-16
were used for the lift coefficient and rolling-moment coefficient, respectively. Newtonβs
method was used for each case as outlined by Phillips [25] with a convergence criterion
of 1.0 Γ 10-12 to satisfy the Jacobian system of equations for the numerical lifting-line
algorithm. For the computations shown here, an airfoil section with a lift slope of 2Ο
21
and a zero-lift angle of attack of 0 were used, which corresponds to a thin airfoil with
zero camber.
3.3 Grid Convergence and Optimization
Multiple grid densities were analyzed to ensure fully grid-converged values from
the numerical LL algorithm. Figure 5 shows results for induced drag as a function of grid
density for several rolling-moment coefficients. Figure 6 shows the induced drag
coefficient, deflection angle, and yawing-moment coefficient predicted by the numerical
LL algorithm as a function of grid density given a rolling-moment coefficient of 0.1. The
values suggest grid convergence is achieved with 80 nodes over the semispan. A grid
density of 100 nodes over the semispan is used in the rest of the analysis to give a preferable
balance of accuracy and computational cost.
Fig. 5 Grid-convergence analysis of induced drag coefficient at several prescribed
rolling-moment coefficients.
The optimum spanwise size and location of ailerons for minimum induced drag is
found by using an open-source optimization algorithm, Optix, created and used by Hodson,
22
et al. [34], available on GitHub2. The algorithm makes use of the Broyden [35], Fletcher
[36], Goldfarb [37], and Shanno [38] (BFGS) method to iteratively find a minimum. For
the aileron root and tip, the optimization algorithm used decimal numbers out to machine
precision at double-precision computing. With a grid density of 100 nodes and small
changes to the span of the aileron section, the number of nodes assigned to the aileron
would change, changing the induced drag and presenting a similar convergence challenge
as the classical LL theory. To counter this issue, the optimization algorithm used two loops.
The inner loop could exclusively change the aileron geometry, while the outer loop could
exclusively change the redistribution of nodes so that the number of nodes assigned to a
section of the wing would remain proportional to the span of the section. In other words,
an aileron making up 50% of the semispan would contain 50 nodes after redistribution.
Fig. 6 Aircraft properties as a function of grid density.
2 https://github.com/usuaero/Optix
23
Figures 7 and 8 show induced drag contours for a wing with a prescribed rolling-
moment coefficient, lift coefficient, and aspect ratio. A diagonal line boundary defines the
limiting case of an infinitely small aileron at any location in the semispan. The (x,y)
coordinates on this plot represent the beginning spanwise location (x) and the ending
spanwise location (y) of the aileron. Contour lines extend to aileron deflections past 25
degrees to show data trends, even though this study only analyzes the inviscid case and
does not consider stall characteristics. Aileron deflections more than 25 degrees are
typically not practical for aircraft design. A circle at the top of the plots shows the minimum
induced drag, with the corresponding value shown in text at the bottom right. In Fig. 7,
which shows induced drag contours with a prescribed rolling-moment coefficient of 0.04,
lift coefficient of 0.5, and aspect ratio of 8, a low gradient near the minimum allows for
movement of the aileron tip between 10 and 40 percent of the wing semispan with less than
a 2% increase in induced drag above the minimum. Figure 8 changes the prescribed rolling-
moment coefficient to 0.1 while keeping all other parameters the same as Fig. 7, which
shows an increased sensitivity of the induced drag to aileron size and position based on
prescribed rolling moment. However, movement of the aileron tip between 10 and 40
percent of the wing semispan only increases the induced drag less than 4% above the
minimum.
24
Fig. 7 Contour of induced drag coefficient at a rolling-moment coefficient of 0.04.
Fig. 8 Contour of induced drag coefficient at a rolling-moment coefficient of 0.1.
Figures 9 and 10 show the contour plots for deflection angle and yawing moment,
respectively, matching the parameters used in Fig. 8. For the deflection angle, the minimum
is found at the point (0,100) in the plot, indicating an aileron extending from the wing root
to the wing tip. For the yawing moment, the minimum can be found by making the aileron
span small and close to the wing root. These minimums are consistently at the same edges
of the domain with each specified rolling moment, lift, aspect ratio, and taper ratio.
25
Fig. 9 Contour plot of deflection angle (in degrees).
Fig. 10 Contour plot of yawing moment coefficient.
Regardless of rolling moment or lift, the minimum induced drag in Figs. 7 and 8 is
found where the aileron tip meets the wing tip. Gradient-based optimization techniques
produce difficulties when the optimum is close to a boundary. With these difficulties in
mind, the aileron tip was limited to coincide with the wing tip for the following analysis,
26
or πΏπ‘ = 1. In other words, the wing section containing the wing tip is infinitely small, and
only the aileron root location could vary to minimize induced drag.
27
CHAPTER 4
EMPIRICAL RELATIONS FOR DESIGN BASED ON RESULTS
4.1 Processing the Results
From Eq. (43)(37), it is evident that the increase in induced drag due to aileron
deflection is dependent on π π·β. From Eq. (51), it is evident that the corresponding yawing
moment depends on π π in the absence of symmetric twist. Equations (13), (44), and (52)
show that π π·β and π π are functions of the wing planform as well as the aileron geometry
because of their dependence on the decomposed Fourier coefficient ππ. These equations
also show that π π·β is independent of prescribed rolling-moment and lift. As the classical
lifting-line theory presented convergence limitations as shown previously, the numerical
lifting-line algorithm combined with a gradient-based optimization algorithm discussed
above were used to optimize the aileron geometry to minimize induced drag as well as find
the corresponding yawing moment for a range of aileron geometries. By combining the
results from the numerical LL method and Eqs. (53)β(55), empirical relations can be
obtained for the aileron root πΏπ, the rolling-moment factor π π·β, and the yawing-moment
factor π π. Cases were run with aspect ratio varied from 4 to 20 in increments of 2 and taper
ratio varied from 0.0 to 1.0 in increments of 0.01.
4.2 Results
4.2.1 Optimal Aileron Root
A plot of aileron roots that minimize induced drag for various wing planforms are
given in Fig. 11. As aspect ratio and taper ratio increase, the aileron root must move closer
to the root of the wing to achieve the minimum induced drag for the wing. Feifel [17]
28
reported that conventional single-segment ailerons are optimally sized for elliptical wings
at 70% semispan. Referencing Fig. 2, the closest tapered wing planform to an elliptical
planform is about a wing with a taper ratio of 0.4. The aileron root values at π π = 0.4 agree
closely with what Feifel [17] reported.
Fig. 11 Aileron root positions based on aspect ratio and taper ratio to achieve
minimum induced drag.
4.2.2 Values of π π·β
A graph of π π·β for optimal aileron design with changes in aspect ratio and taper
ratio is given in Fig. 12. Looking at Fig. 12 and Eq. (43), results for π π·β can be seen as a
percent increase in the induced drag increase due to aileron geometry versus a wing using
the optimum twist distribution function given in Eq. (33), which would give a π π·β of 0.
Since a variable-continuous trailing-edge wing could be designed to produce the optimum
twist distribution function given in Eq. (33), the results in Fig. 12 give insight into the
advantages of VCCTE and VCCW technology. As an example, a wing using discrete
ailerons with an aspect ratio of 8 and taper ratio of 0.4 would produce 10% more induced
29
drag than a variable-continuous trailing-edge wing. Note from Fig. 12 that the minimum
π π·β for any aspect ratio is found at a taper ratio of 1.0.
Fig. 12 Values of πΏπ«π΅ using optimal aileron design as a function of taper ratio for
aspect ratios ranging from 4 to 20.
4.2.3 Values of π π
A graph of π π is given in Fig. 13 for multiple aspect ratios and taper ratios. Note at
a taper ratio of about 0.32, the value for π π is the same for all aspect ratios. At lower taper
ratios, π π is generally less, which decreases the adverse yawing moment. Note that in Eq.
(51), a π π less than 0 provides proverse yaw. Regardless of the wing planform, Fig. 12
shows that if zero twist is applied to the wing geometry, a wing with optimally-placed
ailerons to minimize induced drag will not produce proverse yaw.
30
Fig. 13 Values of πΏπ using optimal aileron design as a function of taper ratio for
aspect ratios ranging from 4 to 20.
4.3 Application Example
An example of how to design a wing with optimally sized ailerons and find the
induced drag and correlated yawing moment is warranted. A designer with a previously
chosen aspect and taper ratio, in this example 14 and 0.6 respectively, could look at Fig.
11 and see that setting the aileron root at 26.5% of the wing semispan minimizes the
induced drag from the aileron. Figure 12 gives a corresponding π π·β of about 0.11, while
Fig. 2 gives a π π· of about 0.045. These values can be used along with a desired lift and
rolling moment in Eq. (45) to give the total induced drag for the wing. With a prescribed
lift coefficient of 0.5 and rolling moment coefficient of 0.05, the induced drag coefficient
is 7.99 Γ 10-3. Figure 13 gives a π π value of about 3.42, which can be used along with a
desired lift and rolling moment in Eq. (51) to give the yawing moment for the wing, which
in this case is -1.94 Γ 10-3. This methodology offers excellent estimates for initial wing
design.
An important note is that symmetric twist (washout) has not been considered in this
31
analysis, which would change the values for π π. With symmetric twist, the values of π π
could drop below 0 in certain conditions and the wing could then create proverse yaw
during roll. This is a topic of future work.
32
CHAPTER 5
CONCLUSION
Aileron geometry determines the induced drag produced by an aileron deflection
for a prescribed rolling moment. A potential flow lifting-line optimization for aileron
spanwise size and position can minimize the induced drag increase from ailerons. Prandtl's
classical LL theory is the foundation for this approach and allows for calculation of
spanwise section-lift distribution for any wing planform and twist distribution function. An
optimum normalized spanwise symmetric twist distribution function minimizes induced
drag for a symmetric wing planform in steady level flight, while an antisymmetric twist
distribution function minimizes induced drag for any symmetric wing planform and
prescribed rolling moment with zero rolling rate. Ailerons will typically produce more
induced drag than the optimum twist solution to produce a rolling moment. Results from
this theory show how the induced drag and yawing moment are related to planform, aileron
design, lift, and rolling moment. For this study, the yawing moment calculations neglected
symmetric wing twist (washout) effects.
This optimization made use of a numerical LL algorithm, as the classical lifting-
line theory had grid-clustering limitations. Based on the grid resolution and convergence
for the induced drag, deflection angle, and yawing moment, 100 nodes per semispan were
used in this analysis. A gradient-based optimization technique is used to find the aileron
spanwise size and position for minimum induced drag. The aileron tip for minimum
induced drag is found to meet the wing tip position, so the optimization was limited to
changing the aileron root while the aileron tip was locked to the wing tip. The optimization
produced results for aileron root positions for minimum induced drag, as well as
33
coefficients for finding the corresponding minimum induced drag and yawing moment.
Coefficients for finding the induced drag include π π· and π π·β , where π π· can be
viewed as an increase in drag due to a deviation in wing planform from an elliptic wing,
and π π·β can be viewed as a percent increase in induced drag due to a deviation from the
optimal twist distribution function through using aileron design. Coefficients for finding
the corresponding yawing moment include π π , where π π can be viewed as a
proportionality constant for adverse yaw. The minimum increase in induced drag from
ailerons is found with a taper ratio of 1.0, while the least adverse yaw can be found with a
taper ratio of 0. Regardless of the wing planform, if zero twist is applied to the wing
geometry, a wing with optimally-placed ailerons to minimize induced drag will not produce
proverse yaw.
Optimal aileron root results can be used in initial design work for aileron geometries
to minimize induced drag. Although optimal aileron solutions for induced drag in all cases
produce adverse yaw, theory suggests that wing symmetric twist (washout) can be used
such that aileron deflection would produce proverse yaw. This is a topic of future research.
37
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